As long as you are not referring to QFN packages, but rather the PLCC and QFP styles of package, there are test clips for complete 4-sided access. I have some here up to 200something pins. The larger pitches like 0.65 and 0.8 are rather affordable and I have several kits of them.
But my favorite solution for the 0.5 and 0.65mm pitches, which is working very well as long as you do not need all pins at once are the so-called WEDGE PROBES. They are available in 2 to 16 pins and with a sensible combination of them you can run most tests that you'll ever need to.
They are surprisingly robust and offer good signal integrity. Tek has another name for them.
For the two-sided packages, complete clips are not only available for SOC, but also for the various TSOP and MSOP packages. It is just QFN which leaves you not much choice except a positioner.
Names to look for (besides HPAK and Tek): Pomona, AP Products, Delta Probe, Emulation Technology, Winslow Adaptics
Some pics should illustrate the mentioned versions..